Question

How do you solve \[{x^2} - 10x - 11 = 0\]?

Answer 1

Hint: Quadratic factorization using splitting of middle term: In this method splitting of middle term into two factors.
Quadratic Factorization using Splitting of Middle Term which is \[x\] term is the sum of two factors and product equal to last term.
First, we will factor 11. So, we can write \[11 = 11 \times 1\]
Since, there is a subtraction sign in front of 11, we will use the factors of 11 and the subtraction sign to make it 10.

Complete step by step answer:
It is given that; \[{x^2} - 10x - 11 = 0\]
We have to solve the equation.
To solve the equation, we will apply a middle term factorization method.
We know that, \[11 = 11 \times 1\]
So, we can write the equation as,
\[ \Rightarrow {x^2} - (11 - 1)x - 11 = 0\]
Simplifying we get,
\[ \Rightarrow {x^2} - 11x + x - 11 = 0\]
Simplifying again we get,
\[ \Rightarrow x(x - 11) + 1(x - 11) = 0\]
Simplifying we get,
\[ \Rightarrow (x - 11)(x + 1) = 0\]
We know that, when the product of two or more terms is equal to zero, then each of the terms also equals to zero individually.
So, we have,
\[(x - 11) = 0\] and \[(x + 1) = 0\]
Solving we get,
\[x = 11\] and \[x =- 1\]

Hence, the solution of \[{x^2} - 10x - 11 = 0\] is \[-1,11\].

Note: We can solve the equation by Sreedhar Acharya formula also.
We know that Sridhar wrote down rules for Solving Quadratic equations, hence the most common method of finding the roots of the quadratic equation is recognised as Sridhar Acharya rule.
Consider a Quadratic equation
\[a{x^2} + bx + c = 0\]
where \[x\] is an unknown variable.
\[a,b,c\]are numerical coefficients
Here,
\[a \ne 0\]
The formula to find the roots of this equation will be
\[ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Here, the solution of the given equation \[{x^2} - 10x - 11 = 0\] is,
\[ \Rightarrow x = \dfrac{{10 \pm \sqrt {{{10}^2} + 4 \times 10 \times 11} }}{2}\]
Simplifying we get,
\[ \Rightarrow x = \dfrac{{10 \pm \sqrt {{{10}^2} + 4 \times 1 \times 11} }}{2}\]
Solving we get,
\[ \Rightarrow x =- 1,11\]
Hence, the solution of \[{x^2} - 10x - 11 = 0\] is \[-1,11\].